Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $y \neq 0$. $a = \dfrac{y}{9y - 81} \div \dfrac{8}{y - 9} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{y}{9y - 81} \times \dfrac{y - 9}{8} $ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ y \times (y - 9) } { (9y - 81) \times 8 } $ $ a = \dfrac {y (y - 9)} {8 \times 9(y - 9)} $ $ a = \dfrac{y(y - 9)}{72(y - 9)} $ We can cancel the $y - 9$ so long as $y - 9 \neq 0$ Therefore $y \neq 9$ $a = \dfrac{y \cancel{(y - 9})}{72 \cancel{(y - 9)}} = \dfrac{y}{72} $